Diffusion Approximations for a Class of Sequential Testing Problems
This work addresses decision-making under uncertainty for applications like product assortment selection, though it is incremental as it builds on existing Bayesian and diffusion methods.
The authors tackled the problem of sequential decision-making under uncertainty by deriving a diffusion approximation for Bayesian sequential experimentation, showing that the problem's complexity grows only quadratically with the number of actions. They applied this to assortment selection in e-commerce, using numerical analysis to demonstrate the value of learning and the effectiveness of derived heuristics.
We consider a decision maker who must choose an action in order to maximize a reward function that depends also on an unknown parameter Θ. The decision maker can delay taking the action in order to experiment and gather additional information on Θ. We model the decision maker's problem using a Bayesian sequential experimentation framework and use dynamic programming and diffusion-asymptotic analysis to solve it. For that, we scale our problem in a way that both the average number of experiments that is conducted per unit of time is large and the informativeness of each individual experiment is low. Under such regime, we derive a diffusion approximation for the sequential experimentation problem, which provides a number of important insights about the nature of the problem and its solution. Our solution method also shows that the complexity of the problem grows only quadratically with the cardinality of the set of actions from which the decision maker can choose. We illustrate our methodology and results using a concrete application in the context of assortment selection and new product introduction. Specifically, we study the problem of a seller who wants to select an optimal assortment of products to launch into the marketplace and is uncertain about consumers' preferences. Motivated by emerging practices in e-commerce, we assume that the seller is able to use a crowdvoting system to learn these preferences before a final assortment decision is made. In this context, we undertake an extensive numerical analysis to assess the value of learning and demonstrate the effectiveness and robustness of the heuristics derived from the diffusion approximation.